Direct product
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one talks about the product in category theory, which formalizes these notions.
Examples are the product of sets, groups, rings, and other algebraic structures. The product of topological spaces is another instance.
There is also the direct sum – in some areas this is used interchangeably, while in others it is a different concept.
Examples
- If we think of as the set of real numbers, then the direct product is just the Cartesian product.
- If we think of as the group of real numbers under addition, then the direct product still has as its underlying set. The difference between this and the preceding example is that is now a group, and so we have to also say how to add their elements. This is done by defining.
- If we think of as the ring of real numbers, then the direct product again has as its underlying set. The ring structure ring consists of addition defined by and multiplication defined by.
- However, if we think of as the field of real numbers, then the direct product does not exist – naively defining addition and multiplication componentwise as in the [|above] example would not result in a field since the element does not have a multiplicative inverse.
Group direct product
In group theory one can define the direct product of two groups and, denoted by G × H. For abelian groups which are written additively, it may also be called the direct sum of two groups, denoted by.It is defined as follows:
- the set of the elements of the new group is the Cartesian product of the sets of elements of G and H, that is ;
- on these elements put an operation, defined element-wise:
- : × =
This construction gives a new group. It has a normal subgroup isomorphic to G, and one isomorphic to H.
The reverse also holds, there is the following recognition theorem: If a group K contains two normal subgroups G and H, such that K= GH and the intersection of G and H contains only the identity, then K is isomorphic to G × H. A relaxation of these conditions, requiring only one subgroup to be normal, gives the semidirect product.
As an example, take as G and H two copies of the unique group of order 2, C2: say and. Then C2×C2 =, with the operation element by element. For instance, * = =, and * = =.
With a direct product, we get some natural group homomorphisms for free: the projection maps define by
called the coordinate functions.
Also, every homomorphism f to the direct product is totally determined by its component functions
For any group and any integer n ≥ 0, repeated application of the direct product gives the group of all n-tuples Gn, for example Zn and Rn.
Direct product of modules
The direct product for modules is very similar to the one defined for groups above, using the Cartesian product with the operation of addition being componentwise, and the scalar multiplication just distributing over all the components. Starting from R we get Euclidean space Rn, the prototypical example of a real n-dimensional vector space. The direct product of Rm and Rn is Rm+n.Note that a direct product for a finite index is identical to the direct sum. The direct sum and direct product differ only for infinite indices, where the elements of a direct sum are zero for all but for a finite number of entries. They are dual in the sense of category theory: the direct sum is the coproduct, while the direct product is the product.
For example, consider and, the infinite direct product and direct sum of the real numbers. Only sequences with a finite number of non-zero elements are in Y. For example, is in Y but is not. Both of these sequences are in the direct product X; in fact, Y is a proper subset of X.
Topological space direct product
The direct product for a collection of topological spaces Xi for i in I, some index set, once again makes use of the Cartesian productDefining the topology is a little tricky. For finitely many factors, this is the obvious and natural thing to do: simply take as a basis of open sets to be the collection of all Cartesian products of open subsets from each factor:
This topology is called the product topology. For example, directly defining the product topology on R2 by the open sets of R, the basis for this topology would consist of all disjoint unions of open rectangles in the plane.
The product topology for infinite products has a twist, and this has to do with being able to make all the projection maps continuous and to make all functions into the product continuous if and only if all its component functions are continuous : we take as a basis of open sets to be the collection of all Cartesian products of open subsets from each factor, as before, with the proviso that all but finitely many of the open subsets are the entire factor:
The more natural-sounding topology would be, in this case, to take products of infinitely many open subsets as before, and this does yield a somewhat interesting topology, the box topology. However it is not too difficult to find an example of bunch of continuous component functions whose product function is not continuous. The problem which makes the twist necessary is ultimately rooted in the fact that the intersection of open sets is only guaranteed to be open for finitely many sets in the definition of topology.
Products are nice with respect to preserving properties of their factors; for example, the product of Hausdorff spaces is Hausdorff; the product of connected spaces is connected, and the product of compact spaces is compact. That last one, called Tychonoff's theorem, is yet another equivalence to the axiom of choice.
For more properties and equivalent formulations, see the separate entry product topology.
Direct product of binary relations
On the Cartesian product of two sets with binary relations R and S, define T as aRc and bSd. If R and S are both reflexive, irreflexive, transitive, symmetric, or antisymmetric, then T will be also. Combining properties it follows that this also applies for being a preorder and being an equivalence relation. However if R and S are total relations, T is in not general total.Direct product in universal algebra
If Σ is a fixed signature, I is an arbitrary index set, and i∈I is an indexed family of Σ algebras, the direct product A = ∏i∈I Ai is a Σ algebra defined as follows:- The universe set A of A is the Cartesian product of the universe sets Ai of Ai, formally: A = ∏i∈I Ai;
- For each n and each n-ary operation symbol, its interpretation fA in A is defined componentwise, formally: for all and each, the ith component of is defined as.
As a special case, if the index set the direct product of two Σ algebras A1 and A2 is obtained, written as. If Σ just contains one binary operation f, the above definition of the direct product of groups is obtained, using the notation,,,, and. Similarly, the definition of the direct product of modules is subsumed here.
Categorical product
The direct product can be abstracted to an arbitrary category. In a general category, given a collection of objects Ai and a collection of morphisms pi from A to Ai with i ranging in some index set I, an object A is said to be a categorical product in the category if, for any object B and any collection of morphisms fi from B to Ai, there exists a unique morphism f from B to A such that fi = pi f and this object A is unique. This not only works for two factors, but arbitrarily many.For groups we similarly define the direct product of a more general, arbitrary collection of groups Gi for i in I, I an index set. Denoting the Cartesian product of the groups by G we define multiplication on G with the operation of componentwise multiplication; and corresponding to the pi in the definition above are the projection maps
the functions that take to its ith component gi.